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Proof that the adjoint representation is an endomorphism

  1. Nov 25, 2014 #1
    1. The problem statement, all variables and given/known data
    My textbooks takes for granted that, given a Lie group ##g## and its algebra ##\mathfrak{g}##, we have that ##AXA^{-1} \in \mathfrak{g}##.

    2. Relevant equations
    For ##Y## to be in ##\mathfrak{g}## means that ##e^{tY} \in G## for each ##t \in \mathbf{R}##

    3. The attempt at a solution
    I tried to expand ##\exp(tAXA^{-1})## but I am stuck, since the term inside the exponential is a mix of group and algebra matrices and I don't know how to deal with it.
     
  2. jcsd
  3. Nov 25, 2014 #2

    Orodruin

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    What can you say in general for a function ##f(AXA^{-1})## assuming it can be Taylor expanded?

    Edit: Or rather, what is ##(A X A^{-1})^n##?
     
  4. Nov 25, 2014 #3
    Thanks! So:
    ##\exp (tAXA^{-1}) = 1 + tAXA^{-1} + \frac{t^2}{2}AX^2A^{-1} + ... = A(\exp{tX})A^{-1} \in G##

    Why I was stuck, I don't understand... Thank you for the hint!
     
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